Encoding and Construction of Quantum Codes from $(γ,Δ)$-cyclic Codes over a Class of Non-chain Rings
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910841599688704 |
|---|---|
| author | Prakash, Om Patel, Shikha Islam, Habibul |
| author_facet | Prakash, Om Patel, Shikha Islam, Habibul |
| contents | Let $\mathbb{F}_q$ be a finite field of $q=p^m$ elements where $p$ is a prime and $m$ is a positive integer. This paper considers $(γ,Δ)$-cyclic codes over a class of finite non-chain commutative rings $\mathscr{R}_{q,s}=\mathbb{F}_q[v_1,v_2,\dots,v_s]/\langle v_i-v_i^2,v_iv_j=v_jv_i=0\rangle$ where $γ$ is an automorphism of $\mathscr{R}_{q,s}$, $Δ$ is a $γ$-derivation of $\mathscr{R}_{q,s}$ and $1\leq i\neq j\leq s$ for a positive integer $s$. Here, we show that a $(γ,Δ)$-cyclic code of length $n$ over $\mathscr{R}_{q,s}$ is the direct sum of $(θ,\Im)$-cyclic codes of length $n$ over $\mathbb{F}_q$, where $θ$ is an automorphism of $\mathbb{F}_q$ and $\Im$ is a $θ$-derivation of $\mathbb{F}_q$. Further, necessary and sufficient conditions for both $(γ,Δ)$-cyclic and $(θ,\Im)$-cyclic codes to contain their Euclidean duals are established. Then, we obtain many quantum codes by applying the dual containing criterion on the Gray images of these codes. These codes have better parameters than those available in the literature. Finally, the encoding and error-correction procedures for our proposed quantum codes are discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01904 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Encoding and Construction of Quantum Codes from $(γ,Δ)$-cyclic Codes over a Class of Non-chain Rings Prakash, Om Patel, Shikha Islam, Habibul Information Theory 12L05, 16Z05, 94B05, 94B35, 94B15 Let $\mathbb{F}_q$ be a finite field of $q=p^m$ elements where $p$ is a prime and $m$ is a positive integer. This paper considers $(γ,Δ)$-cyclic codes over a class of finite non-chain commutative rings $\mathscr{R}_{q,s}=\mathbb{F}_q[v_1,v_2,\dots,v_s]/\langle v_i-v_i^2,v_iv_j=v_jv_i=0\rangle$ where $γ$ is an automorphism of $\mathscr{R}_{q,s}$, $Δ$ is a $γ$-derivation of $\mathscr{R}_{q,s}$ and $1\leq i\neq j\leq s$ for a positive integer $s$. Here, we show that a $(γ,Δ)$-cyclic code of length $n$ over $\mathscr{R}_{q,s}$ is the direct sum of $(θ,\Im)$-cyclic codes of length $n$ over $\mathbb{F}_q$, where $θ$ is an automorphism of $\mathbb{F}_q$ and $\Im$ is a $θ$-derivation of $\mathbb{F}_q$. Further, necessary and sufficient conditions for both $(γ,Δ)$-cyclic and $(θ,\Im)$-cyclic codes to contain their Euclidean duals are established. Then, we obtain many quantum codes by applying the dual containing criterion on the Gray images of these codes. These codes have better parameters than those available in the literature. Finally, the encoding and error-correction procedures for our proposed quantum codes are discussed. |
| title | Encoding and Construction of Quantum Codes from $(γ,Δ)$-cyclic Codes over a Class of Non-chain Rings |
| topic | Information Theory 12L05, 16Z05, 94B05, 94B35, 94B15 |
| url | https://arxiv.org/abs/2404.01904 |